# sequences & series

** Quick links**:

► downloadable teaching materials for sequences & series

► syllabus content for the Algebra Topic:

**SL syllabus**(see syllabus section 1.1);

**HL syllabus**(see syllabus section 1.1).

__challenge__: What is the missing last number in the following sequence?

\(10,\;\;11,\;\;12,\;\;13,\;\;14,\;\;20,\;\;22,\;\;101,\;\;?\)

[click on 'eye' for answer]

Although it's interesting (and I would say worthwhile) to discuss sequences that are defined by recursive (or iterative) formulas - such as the Fibonacci sequence - the fact is that the only sequences/series indicated in either the SL or HL syllabus are **arithmetic **and **geometric** **sequences/series** which have explicit (or closed form) formulas.

Answering questions on sequences & series often relies on applying an appropriate formula. Since students have access to the appropriate **formula booklet** (SL or HL) at all times they do not need to memorize the formulas. I do strongly encourage students to state the formulas "in words" rather than simply reciting symbols. For example, the formula should be read as "*the nth term of an arithmetic sequence is equal to the first term plus the product of the # of 'gaps' between terms and the common difference.*" I think that this promotes greater understanding of precisely what the formula does and encourages students to communicate mathematical expressions clearly.

#### Limits and Convergence

These two concepts - **limit **of a function, and **convergence **of a series - are a couple of the more challenging concepts to illustrate and explain. Fortunately, in both SL and HL, the wording in the syllabus is "informal ideas of limit and convergence." This statement is found in section 6.1 of both syllabi in the lead up to the limit definition of the derivative. The most appropriate time to first discuss, and explore, ideas of limits and convergence is with infinite series. Students may accept the formula for the sum of an infinite geometric series given that \(\left| r \right| < 1\), and they may even understand the proof of this formula; but they usually are not shown (informally) that the defining feature of a convergent infinite series is that the limit of the series is the limit of its sequence of partial sums. This does not need to be approached formally as it is in the HL Calculus option topic, but I do think it is worthwhile to spend a few minutes in class demonstrating (again, informally) what the limit of partial sums looks like visually for a couple of infinite series - one that is known to converge and one is known to diverge. It's also nice for students to realize that not all series are either arithmetic or geometric.

Click here to open a Geogebra applet which dynamically shows the sum of * n* terms as

*increases for the*

**n****geometric series**\(\sum\limits_{k = 1}^n {{{\left( {\frac{4}{5}} \right)}^{k - 1}}} \) that converges, and for the infamous

**harmonic series**\(\sum\limits_{k = 1}^n {\frac{1}{k}} \) that diverges. The applet can be used to compare any two series (within reason). One just needs to enter other expressions for \({a_k}\) and \({b_k}\) in the approriate box; and the maximum value of

*can be adjusted also.*

**n**__4 questions__ - ‘accessible’ to ‘discriminating’

__4 questions__-

download: 4_Qs_sequences_series_1_with_answers ; 4_Qs_sequences_series_1_solutions_notes

##### ♦ teaching materials

EXS_1-1-20v2_SLHL_sequences_series

10 exercises on arithmetic & geometric sequences & series (answers included); corrected version 2

EXS_1-1-40v1_HL_seq_series_binomial_theorem

sequences, series & binomial theorem - exercise set for HL (answers included)

EXS_1-1-50v1_SLHL_sequences_series

10 exercises on sequences & series for SL and HL (answers included)

WRK_1-1-40v2_SLHL_series_sigma_notation

worksheet for arithmetic & geometric series, along with sigma notation; 3 worked examples and 3 exercises with worked solutions

WRK_1-1-50v1_SLHL_sequences_series worksheet for arithmetic & geometric series (good follow-up to previous worksheet 1-1-40 above); 4 worked examples and 6 exercises with answers

Quiz_HL_seq_series_v1

Quiz with 5 questions - 3 with GDC allowed, and 2 with no GDC. Quiz can work for SL but the questions are a bit more *moderate *to *discrimating *in difficulty level. **Worked solutions** available below.

Quiz_HL_seq_series_v1_SOL_KEY **Worked solutions** for the HL sequences & series Quiz above

Test_SL_seq_series_v1

SL Test - sequences & series; 9 questions - GDC allowed on all questions (**worked solutions** below)

Test_SL_seq_series_v1_SOL_KEY **worked solutions** for all 9 questions on SL Test - sequences & series (above)

Test2_SL_seq_series_bin_thm_v1

SL Test - sequences & series, binomial theorem, sigma notation; 9 questions - no GDC on 4 questions, GDC allowed on 5 questions (**worked solutions** below)

Test2_SL_seq_series_bin_thm_v1_SOL_KEY **worked solutions** for SL Test on sequences & series, binomial theorem and sigma notation above

Test1_HL_seq_series_bin_thm_v1

HL Test #1 - sequences, series, binomial theorem. GDC allowed on all questions. **Answers **on 2nd page. **Worked solutions **available below.

Test1_HL_seq_series_bin_thm_SOL_KEY_v1 **Worked solutions** for HL Test #1 on sequences, series & binomial theorem above.

Test2_HL_seq_series_bin_thm_v1

HL Test #2 - sequences & series, binomial theorem. GDC allowed on all questions. **Worked solutions** available below.

Test2_HL_seq_series_bin_thm_v1_SOL_KEY **Worked solutions** for HL Test #2 on sequences & series, and binomial theorem above.

Test3_HL_seq_series_bin_thm_v1

HL Test #3; content: sequences & series; binomial expansions sigma notation and binomial theorem; 10 questions: 5 with no GDC, and 5 with GDC. **Worked solutions** available below.

Test3_HL_seq_series_bin_thm_v1_SOL_KEY **Worked solutions** for HL Test #3 on sequences & series, binomial expansions, sigma notation, binomial theorem; no GDC and GDC allowed questions